#386787

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(5,6,7,8)(9,10)(11,19)(12,24)(13,29)(14,17)(15,22)(16,27)(18,26)(20,30)(21,32)(23,31)(25,33)(28,38)(34,39)(35,36)(37,40);(2,4)(9,11)(10,15)(12,25)(13,21)(14,23)(16,33)(17,32)(18,31)(19,29)(20,28)(22,26)(24,35)(27,37)(30,34)(36,39)(38,40);(1,2,3,4)(9,12,26,39,14,28,13,27)(10,16,29,38,17,34,18,24)(11,20,22,37,23,25,21,36)(15,30,19,35,32,33,31,40);(10,17)(11,21,23,22)(15,19,32,31)(16,34)(18,29)(20,36,25,37)(24,38)(30,35,33,40)

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293983);